Evaluate the Summation sum from i=1 to 5 of -3*2^i
The problem asks for the evaluation of a summation, which is a mathematical term for adding up a series of numbers generated by a formula. Specifically, you are to calculate the total sum that results from the given expression for each integer value of 'i' starting from 1 and going up to 5. The expression to be used in the summation is '-3*2^i', which means you need to multiply -3 by 2 raised to the power of 'i' (where 'i' takes on the values 1, 2, 3, 4, and 5 in successive calculations), and then add all those individual products together to get the final result.
$\sum_{i = 1}^{5} - 3 \cdot 2^{i}$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the summation for each value of \( i \) from 1 to 5.
\[ -3 \times 2^1, -3 \times 2^2, -3 \times 2^3, -3 \times 2^4, -3 \times 2^5 \]
Step 2: Calculation of the Sum
Compute the sum of the expanded series.
\[ \sum_{i=1}^{5} (-3 \times 2^i) = -3 \times 2^1 - 3 \times 2^2 - 3 \times 2^3 - 3 \times 2^4 - 3 \times 2^5 = -186 \]
Knowledge Notes:
To solve the given problem, we need to understand the concept of summation and the geometric progression.
Summation (危) is a mathematical notation used to represent the addition of a sequence of numbers. The summation notation includes an expression to be summed, a variable that iterates over a set of integers, and limits that define the range of the variable.
In this problem, the summation is from \( i = 1 \) to \( i = 5 \) of the expression \( -3 \cdot 2^i \). This means we need to evaluate the expression for each integer value of \( i \) within the given range and then sum the results.
The expression \( -3 \cdot 2^i \) represents a geometric sequence where each term is obtained by multiplying the previous term by a common ratio, in this case, 2. The first term is \( -3 \cdot 2^1 \) and the common ratio is 2.
To solve the summation, we expand the series by calculating each term separately and then summing them up. This is done by substituting \( i \) with each integer from 1 to 5 in the expression \( -3 \cdot 2^i \) and adding the results.
The final step is to simplify the sum of the expanded series to get the final result.
By understanding these concepts, we can solve the summation problem methodically and arrive at the correct answer.
